Global solutions and asymptotic behavior for two dimensional gravity water waves

By Thomas Alazard, Jean-Marc Delort. 2013

180 citations recorded by the source corpus.

This paper is devoted to the proof of a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds. The proof is based on a bootstrap argument involving $L^2$ and $L^\infty$ estimates. The $L^2$ bounds are proved in a companion paper of this article. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation of the water waves equation. We give here the proof of the uniform bounds, interpreting the equation in a semi-classical way, and combining Klainerman vector fields with the description of the solution in terms of semi-classical lagrangian distributions. This, together with the $L^2$ estimates of the companion paper, allows us to deduce our main global existence result.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2b0b6cce8866e14b070bc2685a291ff78836f3c8

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Global solutions for the gravity water waves system in 2d (2013, result): “e anonymous referees for carefully reading the manuscript and for their many useful comments. After submission of this manuscript a different proof of a similar result was given by Alazard and Delort [AD13]. The ﬁrst autho r was partially supported by a Packard Fellowship and NSF Grant DMS 1265818. ”
MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, background): “tised by (r;’) is the following: ’_ = (1 2r)(1 + 2cos2’); r_ = 4r(1 r)sin2’; (4.11) which is also Hamiltonian with energy h(’;r) = r(1 r)[1 + 2cos(2’)]: Due to our mass normalization, we have that r2[0;1] for all time. Notice13 that (I 0;I 1; 0; 1) can all be derived from the knowledge of (’;r) and ”
Time quasi-periodic gravity water waves in finite depth (2017, methodology): “d = 1) it has been proved by Alazard-Delort [4] and Ionescu-Pusateri [36].”
Two Dimensional Water Waves in Holomorphic Coordinates (2014, unknown): “Such “quadratic” quasilinear energy functionals appear in most recent articles dealing with the local well-posedness problem, see [1,3,15,26].”
Two dimensional water waves in holomorphic coordinates II: global solutions (2014, unknown): “A similar asymptotic profile, but in Eulerian coordinates, was obtained by Alazard and Delort in [3].”